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G = C84.C4order 336 = 24·3·7

1st non-split extension by C84 of C4 acting via C4/C2=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C84.1C4, C4.Dic21, C28.50D6, C4.15D42, C217M4(2), C12.51D14, C28.1Dic3, C12.1Dic7, C22.Dic21, C84.57C22, C21⋊C85C2, (C2×C42).3C4, (C2×C28).5S3, (C2×C84).7C2, (C2×C12).5D7, (C2×C4).2D21, C42.30(C2×C4), C32(C4.Dic7), C72(C4.Dic3), (C2×C6).3Dic7, C6.7(C2×Dic7), C14.7(C2×Dic3), C2.3(C2×Dic21), (C2×C14).3Dic3, SmallGroup(336,96)

Series: Derived Chief Lower central Upper central

C1C42 — C84.C4
C1C7C21C42C84C21⋊C8 — C84.C4
C21C42 — C84.C4
C1C4C2×C4

Generators and relations for C84.C4
 G = < a,b | a84=1, b4=a42, bab-1=a-1 >

2C2
2C6
2C14
21C8
21C8
2C42
21M4(2)
7C3⋊C8
7C3⋊C8
3C7⋊C8
3C7⋊C8
7C4.Dic3
3C4.Dic7

Smallest permutation representation of C84.C4
On 168 points
Generators in S168
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168)
(1 145 22 124 43 103 64 166)(2 144 23 123 44 102 65 165)(3 143 24 122 45 101 66 164)(4 142 25 121 46 100 67 163)(5 141 26 120 47 99 68 162)(6 140 27 119 48 98 69 161)(7 139 28 118 49 97 70 160)(8 138 29 117 50 96 71 159)(9 137 30 116 51 95 72 158)(10 136 31 115 52 94 73 157)(11 135 32 114 53 93 74 156)(12 134 33 113 54 92 75 155)(13 133 34 112 55 91 76 154)(14 132 35 111 56 90 77 153)(15 131 36 110 57 89 78 152)(16 130 37 109 58 88 79 151)(17 129 38 108 59 87 80 150)(18 128 39 107 60 86 81 149)(19 127 40 106 61 85 82 148)(20 126 41 105 62 168 83 147)(21 125 42 104 63 167 84 146)

G:=sub<Sym(168)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168), (1,145,22,124,43,103,64,166)(2,144,23,123,44,102,65,165)(3,143,24,122,45,101,66,164)(4,142,25,121,46,100,67,163)(5,141,26,120,47,99,68,162)(6,140,27,119,48,98,69,161)(7,139,28,118,49,97,70,160)(8,138,29,117,50,96,71,159)(9,137,30,116,51,95,72,158)(10,136,31,115,52,94,73,157)(11,135,32,114,53,93,74,156)(12,134,33,113,54,92,75,155)(13,133,34,112,55,91,76,154)(14,132,35,111,56,90,77,153)(15,131,36,110,57,89,78,152)(16,130,37,109,58,88,79,151)(17,129,38,108,59,87,80,150)(18,128,39,107,60,86,81,149)(19,127,40,106,61,85,82,148)(20,126,41,105,62,168,83,147)(21,125,42,104,63,167,84,146)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168), (1,145,22,124,43,103,64,166)(2,144,23,123,44,102,65,165)(3,143,24,122,45,101,66,164)(4,142,25,121,46,100,67,163)(5,141,26,120,47,99,68,162)(6,140,27,119,48,98,69,161)(7,139,28,118,49,97,70,160)(8,138,29,117,50,96,71,159)(9,137,30,116,51,95,72,158)(10,136,31,115,52,94,73,157)(11,135,32,114,53,93,74,156)(12,134,33,113,54,92,75,155)(13,133,34,112,55,91,76,154)(14,132,35,111,56,90,77,153)(15,131,36,110,57,89,78,152)(16,130,37,109,58,88,79,151)(17,129,38,108,59,87,80,150)(18,128,39,107,60,86,81,149)(19,127,40,106,61,85,82,148)(20,126,41,105,62,168,83,147)(21,125,42,104,63,167,84,146) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168)], [(1,145,22,124,43,103,64,166),(2,144,23,123,44,102,65,165),(3,143,24,122,45,101,66,164),(4,142,25,121,46,100,67,163),(5,141,26,120,47,99,68,162),(6,140,27,119,48,98,69,161),(7,139,28,118,49,97,70,160),(8,138,29,117,50,96,71,159),(9,137,30,116,51,95,72,158),(10,136,31,115,52,94,73,157),(11,135,32,114,53,93,74,156),(12,134,33,113,54,92,75,155),(13,133,34,112,55,91,76,154),(14,132,35,111,56,90,77,153),(15,131,36,110,57,89,78,152),(16,130,37,109,58,88,79,151),(17,129,38,108,59,87,80,150),(18,128,39,107,60,86,81,149),(19,127,40,106,61,85,82,148),(20,126,41,105,62,168,83,147),(21,125,42,104,63,167,84,146)])

90 conjugacy classes

class 1 2A2B 3 4A4B4C6A6B6C7A7B7C8A8B8C8D12A12B12C12D14A···14I21A···21F28A···28L42A···42R84A···84X
order122344466677788881212121214···1421···2128···2842···4284···84
size11221122222224242424222222···22···22···22···22···2

90 irreducible representations

dim111112222222222222222
type++++-+-+-+-+-+-
imageC1C2C2C4C4S3Dic3D6Dic3D7M4(2)Dic7D14Dic7D21C4.Dic3Dic21D42Dic21C4.Dic7C84.C4
kernelC84.C4C21⋊C8C2×C84C84C2×C42C2×C28C28C28C2×C14C2×C12C21C12C12C2×C6C2×C4C7C4C4C22C3C1
# reps12122111132333646661224

Matrix representation of C84.C4 in GL4(𝔽337) generated by

23632100
2824800
0022098
00072
,
1621700
16117500
0031431
007623
G:=sub<GL(4,GF(337))| [236,28,0,0,321,248,0,0,0,0,220,0,0,0,98,72],[162,161,0,0,17,175,0,0,0,0,314,76,0,0,31,23] >;

C84.C4 in GAP, Magma, Sage, TeX

C_{84}.C_4
% in TeX

G:=Group("C84.C4");
// GroupNames label

G:=SmallGroup(336,96);
// by ID

G=gap.SmallGroup(336,96);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-7,24,121,50,964,10373]);
// Polycyclic

G:=Group<a,b|a^84=1,b^4=a^42,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of C84.C4 in TeX

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